Subgroups — #24919516

Order 10624.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128391	C₂	1	1	yes	maximal, characteristic
2	D₄×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
3	#19128070	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128043	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128032	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128009	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127959	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127942	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127912	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127859	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127859	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127809	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127711	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#14581717	C₂	1	1	yes	maximal, characteristic, complemented
15	#435×C₈₃	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₁₆₆	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₆₆	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
20	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
21	#14581792	C₂²	1	1	yes	characteristic
22	#14581791	—	2	1	no	—
23	#14581785	—	2	1	no	—
24	#14581784	C₂²	1	1	yes	characteristic
25	#14581782	—	2	1	no	—
26	#14581773	—	2	1	no	—
27	#14581771	C₂²	1	1	yes	characteristic
28	#14581771	C₂²	1	1	yes	characteristic
29	#14581771	C₂²	1	1	yes	characteristic
30	#14581771	C₂²	1	1	yes	characteristic
31	#14581769	—	2	1	no	—
32	#14581769	—	2	1	no	—
33	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
35	D₄×Dic₈₃	—	2	1	no	—
36	#14581751	C₂²	1	1	yes	characteristic
37	#14581748	—	2	1	no	—
38	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
39	#14581726	C₂²	1	1	yes	characteristic
40	#14581724	C₂²	1	1	yes	characteristic
41	#14581724	C₂²	1	1	yes	characteristic
42	#14581717	C₂²	1	1	yes	characteristic
43	#14581717	C₂²	1	1	yes	characteristic
44	#14581717	C₂²	1	1	yes	characteristic
45	#14581717	C₂²	1	1	yes	characteristic
46	#14581717	C₂²	1	1	yes	characteristic
47	#14581717	C₂²	1	1	yes	characteristic
48	#14581717	C₂²	1	1	yes	characteristic
49	#14581717	C₂²	1	1	yes	characteristic
50	#14581717	C₂²	1	1	yes	characteristic
51	#14581717	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
58	#97×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	2	no	—
60	C₂×Dic₁₆₆	—	2	2	no	—
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×Dic₁₆₆	—	2	1	no	—
63	C₂×Dic₁₆₆	—	4	1	no	—
64	C₂×Dic₁₆₆	—	4	1	no	—
65	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
68	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
69	C₂²×Dic₈₃	—	2	1	no	—
70	C₂²×Dic₈₃	—	2	1	no	—
71	C₂²×Dic₈₃	—	2	1	no	—
72	Dic₃₃₂	—	2	2	no	—
73	Dic₃₃₂	—	4	1	no	—
74	Dic₃₃₂	—	4	1	no	—
75	C₄⋊Dic₈₃	—	2	1	no	—
76	C₄⋊Dic₈₃	—	2	1	no	—
77	C₄⋊Dic₈₃	—	2	1	no	—
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	Dic₈₃⋊C₄	—	2	1	no	—
86	Dic₈₃⋊C₄	—	2	1	no	—
87	Dic₈₃⋊C₄	—	2	1	no	—
88	Dic₈₃⋊C₄	—	2	1	no	—
89	Dic₈₃⋊C₄	—	2	1	no	—
90	C₂²⋊Dic₈₃	—	2	1	no	—
91	C₂²⋊Dic₈₃	—	2	1	no	—
92	C₂²⋊Dic₈₃	—	2	1	no	—
93	C₂²⋊Dic₈₃	—	2	1	no	—
94	C₂²⋊Dic₈₃	—	2	1	no	—
95	C₄×Dic₈₃	—	2	1	no	—
96	C₄×Dic₈₃	—	2	1	no	—
97	C₄×Dic₈₃	—	2	1	no	—
98	C₂×D₄×C₈₃	D₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
101	M₁₆×C₈₃	—	4	1	no	—
102	C₂×C₈×C₈₃	—	4	1	no	—
103	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
108	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
109	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
110	C₄²×C₈₃	C₂³	1	1	yes	characteristic
111	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
113	D₄×C₈₃	—	2	2	no	—
114	C₂×C₄×C₈₃	—	2	1	no	—
115	C₂×C₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	2	1	no	—
118	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
123	C₂×Dic₈₃	—	2	1	no	—
124	C₂×Dic₈₃	—	2	1	no	—
125	C₂×Dic₈₃	—	2	1	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	—	2	1	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	—	2	1	no	—
130	C₂×Dic₈₃	—	2	1	no	—
131	C₂×Dic₈₃	—	4	2	no	—
132	C₂×Dic₈₃	—	4	2	no	—
133	C₂×Dic₈₃	—	4	2	no	—
134	Dic₁₆₆	—	2	1	no	—
135	Dic₁₆₆	—	2	1	no	—
136	Dic₁₆₆	—	2	1	no	—
137	Dic₁₆₆	—	2	1	no	—
138	Dic₁₆₆	—	2	1	no	—
139	Dic₁₆₆	—	2	1	no	—
140	Dic₁₆₆	—	2	1	no	—
141	Dic₁₆₆	—	4	2	no	—
142	Dic₁₆₆	—	4	2	no	—
143	Dic₁₆₆	#41	1	2	yes	—
144	C₈×C₈₃	—	2	1	no	—
145	C₈×C₈₃	—	2	1	no	—
146	C₈×C₈₃	—	2	1	no	—
147	C₈×C₈₃	—	2	1	no	—
148	C₂²×C₈₃	—	2	1	no	—
149	C₂²×C₈₃	—	2	1	no	—
150	C₂²×C₈₃	—	2	1	no	—
151	C₂²×C₈₃	—	2	1	no	—
152	C₂²×C₈₃	—	2	1	no	—
153	C₂²×C₈₃	#137	1	2	yes	—
154	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₈₃	—	2	1	no	—
156	C₄×C₈₃	—	2	1	no	—
157	C₄×C₈₃	—	2	1	no	—
158	C₄×C₈₃	—	2	1	no	—
159	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
160	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
161	Dic₈₃	—	2	2	no	—
162	Dic₈₃	—	4	1	no	—
163	Dic₈₃	—	4	1	no	—
164	Dic₈₃	—	4	1	no	—
165	Dic₈₃	—	4	1	no	—
166	Dic₈₃	—	4	1	no	—
167	Dic₈₃	—	4	1	no	—
168	Dic₈₃	—	4	1	no	—
169	C₂×C₈₃	—	2	2	no	—
170	C₂×C₈₃	—	4	1	no	—
171	C₂×C₈₃	#546	1	1	yes	characteristic
172	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
173	C₂×C₈₃	#578	1	1	yes	characteristic
174	#3312	—	83	1	no	maximal, Sylow
175	C₈₃	#3312	1	1	yes	characteristic, complemented, Sylow
176	#552	—	83	1	no	—
177	D₄×Q₈	—	83	1	no	—
178	#491	—	83	1	no	—
179	#485	—	83	1	no	—
180	#483	—	83	1	no	—
181	#479	—	83	1	no	—
182	#470	—	83	1	no	—
183	#467	—	83	1	no	—
184	#462	—	83	1	no	—
185	#451	—	83	1	no	—
186	#451	—	83	1	no	—
187	#441	—	83	1	no	—
188	#435	D₈₃	1	1	yes	characteristic
189	#419	—	83	1	no	—
190	C₂×#103	—	83	1	no	—
191	C₂²×Q₈	—	83	1	no	—
192	C₂²×Q₈	—	83	1	no	—
193	C₂×Q₁₆	—	83	1	no	—
194	C₂×Q₁₆	—	83	1	no	—
195	C₂×Q₁₆	—	83	1	no	—
196	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
197	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
198	C₄⋊Q₈	—	83	1	no	—
199	C₄⋊Q₈	—	166	1	no	—
200	#126	—	166	1	no	—
201	#125	—	83	1	no	—
202	#125	—	166	1	no	—
203	#123	—	166	1	no	—
204	C₂²⋊Q₈	—	83	1	no	—
205	C₂²⋊Q₈	—	83	1	no	—
206	C₂²⋊Q₈	—	83	1	no	—
207	C₂²⋊Q₈	—	83	1	no	—
208	C₂²⋊Q₈	—	166	1	no	—
209	C₂²⋊Q₈	—	166	1	no	—
210	C₄×Q₈	—	83	1	no	—
211	C₄×Q₈	—	83	1	no	—
212	C₄×D₄	—	166	1	no	—
213	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
214	#117	—	83	1	no	—
215	#117	—	166	1	no	—
216	C₂×C₄⋊C₄	—	83	1	no	—
217	#107	—	83	1	no	—
218	#106	—	83	1	no	—
219	#106	—	83	1	no	—
220	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
221	#103	—	83	1	no	—
222	#103	—	83	1	no	—
223	#103	—	83	1	no	—
224	#103	—	83	1	no	—
225	#103	—	83	1	no	—
226	#103	—	83	1	no	—
227	#103	—	83	1	no	—
228	#103	—	83	1	no	—
229	#103	—	83	1	no	—
230	#103	—	83	1	no	—
231	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
232	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
233	#97	C₂×D₈₃	1	1	yes	characteristic
234	C₂×Q₈	—	332	1	no	—
235	C₂×Q₈	—	332	1	no	—
236	C₂×Q₈	—	83	1	no	—
237	C₂×Q₈	—	83	1	no	—
238	C₂×Q₈	—	83	1	no	—
239	C₂×Q₈	—	83	1	no	—
240	C₂×Q₈	—	166	2	no	—
241	C₂×Q₈	—	166	2	no	—
242	C₂×Q₈	—	166	1	no	—
243	C₂×Q₈	—	166	1	no	—
244	C₂×D₄	D₃₃₂	1	1	yes	characteristic
245	C₂²×C₄	—	166	1	no	—
246	C₂²×C₄	—	166	1	no	—
247	C₂²×C₄	—	166	1	no	—
248	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
250	Q₁₆	—	332	1	no	—
251	Q₁₆	—	332	1	no	—
252	Q₁₆	—	166	2	no	—
253	M₁₆	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
257	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
259	C₄⋊C₄	—	83	1	no	—
260	C₄⋊C₄	—	83	1	no	—
261	C₄⋊C₄	—	83	1	no	—
262	C₄⋊C₄	—	83	1	no	—
263	C₄⋊C₄	—	166	1	no	—
264	C₄⋊C₄	—	166	1	no	—
265	C₄⋊C₄	—	166	1	no	—
266	C₄⋊C₄	—	166	1	no	—
267	C₄⋊C₄	—	166	1	no	—
268	C₄⋊C₄	—	166	1	no	—
269	C₄⋊C₄	—	166	1	no	—
270	C₄⋊C₄	—	166	1	no	—
271	C₄⋊C₄	—	166	1	no	—
272	C₄⋊C₄	—	166	1	no	—
273	C₄⋊C₄	—	166	1	no	—
274	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
275	C₂²⋊C₄	—	166	1	no	—
276	C₂²⋊C₄	—	166	1	no	—
277	C₂²⋊C₄	—	166	1	no	—
278	C₂²⋊C₄	—	166	1	no	—
279	C₂²⋊C₄	—	166	1	no	—
280	C₂²⋊C₄	D₃₃₂	1	1	yes	characteristic
281	C₂²⋊C₄	D₃₃₂	1	1	yes	characteristic
282	C₄²	—	166	1	no	—
283	C₄²	—	166	1	no	—
284	C₄²	—	166	1	no	—
285	C₄²	C₂²×D₈₃	1	1	yes	characteristic
286	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
287	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
288	Q₈	—	332	2	no	—
289	Q₈	—	332	2	no	—
290	Q₈	—	83	2	no	—
291	Q₈	—	166	1	no	—
292	Q₈	—	166	1	no	—
293	Q₈	—	166	1	no	—
294	Q₈	—	166	1	no	—
295	Q₈	—	166	1	no	—
296	Q₈	—	166	1	no	—
297	Q₈	—	166	1	no	—
298	D₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	332	2	no	—
304	C₂×C₄	—	332	2	no	—
305	C₂×C₄	—	332	2	no	—
306	C₂×C₄	—	166	1	no	—
307	C₂×C₄	—	166	1	no	—
308	C₂×C₄	—	166	1	no	—
309	C₂×C₄	—	166	1	no	—
310	C₂×C₄	—	166	1	no	—
311	C₂×C₄	—	166	1	no	—
312	C₂×C₄	—	166	1	no	—
313	C₂×C₄	—	166	1	no	—
314	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
315	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
316	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
317	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
318	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	#14581825	1	2	yes	—
329	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	332	1	no	—
335	C₄	—	332	1	no	—
336	C₄	—	332	1	no	—
337	C₄	—	332	1	no	—
338	C₄	—	332	1	no	—
339	C₄	—	332	1	no	—
340	C₄	—	332	1	no	—
341	C₄	—	166	2	no	—
342	C₄	#41×D₈₃	1	1	yes	characteristic
343	C₄	#14581837	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	4	1	no	—
346	C₂	#19128346	1	1	yes	characteristic
347	C₂	C₂×#14581825	1	1	yes	characteristic
348	C₂	#19128518	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₃₂
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#435×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×D₄×C₈₃	D₄
hypercenter	#435	D₈₃
hypocenter	C₈₃	#3312
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #19128070	C₂	semidirect
4. #19128043	C₂	semidirect
5. #19128032	C₂	semidirect
6. #19128009	C₂	semidirect
7. #19127959	C₂	semidirect
8. #19127942	C₂	semidirect
9. #19127912	C₂	semidirect
10. #19127859	C₂	semidirect
11. #19127859	C₂	semidirect
12. #19127809	C₂	semidirect
13. #19127711	C₂	semidirect
14. C₂×#14581717	C₂	semidirect
175. C₈₃	#3312	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19128391	2
D₄×Dic₁₆₆	2
#19128070	2
#19128043	2
#19128032	2
#19128009	2
#19127959	2
#19127942	2
#19127912	2
#19127859	2
#19127809	2
#19127711	2
C₂×#14581717	2
#435×C₈₃	2
#3312	83